On the curvature of symmetric products of a compact Riemann surface

TL;DR

This paper provides a simple, direct proof that symmetric products of compact Riemann surfaces of genus at least two cannot admit Kähler metrics with nonnegative holomorphic bisectional curvature for certain dimensions.

Contribution

It offers a straightforward proof of a known result regarding the curvature properties of symmetric products of Riemann surfaces, simplifying previous approaches.

Findings

Symmetric products of genus ≥ 2 surfaces lack Kähler metrics with nonnegative holomorphic bisectional curvature for n ≤ 2(genus-1).

The proof simplifies understanding of curvature restrictions on symmetric products.

Confirms the nonexistence of certain curvature conditions on these complex manifolds.

Abstract

Let $X$ be a compact connected Riemann surface of genus at least two. The main theorem of arxiv:1010.1488 says that for any positive integer $n \leq 2({\rm genus}(X)-1)$, the symmetric product $S^n(X)$ does not admit any K\"ahler metric satisfying the condition that all the holomorphic bisectional curvatures are nonnegative. Our aim here is to give a very simple and direct proof of this result of B\"okstedt and Rom\~ao.